Although the ship offers several excursions we would like to go this 'on our own'. Has anyone done gone to this village on their own while in port?

How long does it take to get to Taormina from where the ship docks? How far is the train station from the ship?

I got the information below off the Internet but wonder about the logistics of doing this on our own.
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Taormina's rail station is by the sea, far below the town. It also serves the coastal settlement of Giardini Naxos, and the station is therefore named Taormina - Giardini. The station is on the railway line that runs between Messina and Siracusa, and is also served by trains from Rome.

Buses from the station to Taormina are approximately half-hourly and stop right outside the station, where you will find a timetable. Tickets cost €1.30 and can be bought from the driver, or at the station newsagents (edicola). The zig-zagging journey up to Taormina takes about ten minutes. In Taormina buses stop at the 'Bus Terminal' on Via Luigi Pirandello, somewhat below the town centre. A taxi for the same journey will cost around €12, and will save you from lugging your cases around the town's sloping streets.

The bus company Interbus also operate longer-distance services, to Catania Airport and further afield. Within Taormina, small orange local buses wind their way in a circular route around around the residential periphery of town.
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In 2000 we did Taormina "on our own" thru the ship. There was a person on the bus who gave us some idea what there was to see and do - but we were on our own when we arrived. Bus let us off at foot and we hoofed it up to the village. Don't miss the collesum (sp?) - charge to get in - but views are spectacular. Can see Mt. Etna if weather cooperates. Be sure to walk to the very back of the village and enjoy the views from there. Lots of lovely side streets with wonderful al fresco dining. We were there several hours. I wouldn't want the hassle of trying to arrange public transport. I'd never, ever book a tour of Taormina. It's too easy to see on your own once you arrive.